Average Error: 37.3 → 0.4
Time: 6.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)
double code(double x, double eps) {
	return (sin((x + eps)) - sin(x));
}
↓
double code(double x, double eps) {
	return (fma(sin(x), cbrt(pow((cos(eps) - 1.0), 3.0)), (cos(x) * sin(eps))) + fma(-sin(x), 1.0, (sin(x) * 1.0)));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target15.3
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.3

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  2. Using strategy rm

  3. Applied sin-sum21.9

    \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]

  4. Applied associate--l+21.9

    \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity21.9

    \[\leadsto \sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \color{blue}{1 \cdot \sin x}\right)\]

  7. Applied prod-diff21.9

    \[\leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\left(\mathsf{fma}\left(\cos x, \sin \varepsilon, -\sin x \cdot 1\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)\right)}\]

  8. Applied associate-+r+21.9

    \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \mathsf{fma}\left(\cos x, \sin \varepsilon, -\sin x \cdot 1\right)\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)}\]

  9. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \cos x \cdot \sin \varepsilon\right)} + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)\]
  10. Using strategy rm

  11. Applied add-cbrt-cube0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \color{blue}{\sqrt[3]{\left(\left(\cos \varepsilon - 1\right) \cdot \left(\cos \varepsilon - 1\right)\right) \cdot \left(\cos \varepsilon - 1\right)}}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)\]

  12. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \sqrt[3]{\color{blue}{{\left(\cos \varepsilon - 1\right)}^{3}}}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)\]

  13. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))